Topological solitons of \((2 + 1)\) -dimensional field models
[1]}, [2]}, [3]} play an important role in field theory,
high-energy physics, condensed matter physics, cosmology, and hydrodynamics.
Of these, the vortex solutions of the effective theory of superconductivity
[4]} and those of the \(\left(2+1\right)\) -dimensional Abelian Higgs model
[5]} are notable.
Another important example is provided by the soliton solutions of the
\((2+1)\) -dimensional nonlinear \(O\left(3\right)\)  \(\sigma \) -model [6]}.
The invariance of the static energy functional of the nonlinear \(O\left(3\right)\)  \(\sigma \) -model under scale transformations results in a corresponding
zero mode of the quadratic fluctuation operator in the functional neighborhood
of a given soliton solution.
As a consequence, the \(\sigma \) -model possesses a one-parameter family of
soliton solutions with the same energy but different sizes, rather than a
soliton solution of fixed size.
